EEE4176 Applications of Digital Signal Processing Fall 2011 Assistant Prof. Yangmo Yoo Dept. of Electronic Engineering Sogang University

Lecture 4: Filter Banks

Filter Banks Subband processing Spliting the freq range into M subbands Filter bank : Analysis filter bank Process each subband independently : subband coding, etc. Recombine the processed results Synthesis filter bank Perfect reconstruction property No aliasing error No amplitude distortion No phase distortion Quantization error can not be compensated.

Filter Banks Uniform filter bank : All subbands have the same bandwidth zeroth subband Subbands are often overlapped. m-th analysis and synthesis filters usually occupy the same frequency range for Perfect reconstruction. Complex filter vs. real filter : # of subbands are same.

Filter Banks Maximally decimated filter bank Each subband has a bandwidth of (Uniform filter bank case) Each subband signals can be decimated by a factor of Even when the signals are decimated, we still need the perfect reconstruction property. That is, first aliasing error must be eliminated.   Analysis and Synthesis filters are to be properly designed to meet the PR condition

Filter Banks Two-channel Filter Bank

Filter Banks Two-channel Filter Bank Aliasing Term: Alias free condition : LPF : HPF

Filter Banks Two-channel Filter Bank Alias free condition Note: If : Perfect reconstruction Amplitude distortion if Phase distortion if is not linear

Filter Banks Quadrature Mirror Filter (QMF) Bank Simple alias-free condition Transfer function of a QMF bank : is the mirror of around that is, at the quadrature frequency ( : LPF : HPF

Filter Banks Quadrature Mirror Filter (QMF) Bank Note FIR QMF with simple alias-free condition can’t have sharp response If has linear phase, then no phase distortion produced. should be odd. otherwise (severe amplitude distortion) Amplitude distortion By using proper IIR filters, we can eliminate aliasing error, amplitude distortion but not phase distortion. : Linear phase can be minimized using some filter design techniques (ex: Jonston's)

Filter Banks Quadrature Mirror Filter (QMF) Bank Remarks:Linear phase QMF filter bank where No ALD (Alias distortion No PHD --> No AMD but trivial filter So, try to minimize AMD using nontrivial filters Johnston's technique to design filters optimally satisfying Filter design using CAD technique Objective function: : Power symmetry property If we don't stick to , we can achieve real PR FIR QMF

Perfect Reconstruction (PR) Filter Banks Conjugate quadrature filters (CQF) For Alias cancellation, For Perfect reconstruction, : Power symmetric condition

Perfect Reconstruction (PR) Filter Banks Conjugate quadrature filters (CQF) Note: How to design Power Symmetric Filters ? Design : Half band filter Spectral factorization to obtain from for all

Perfect Reconstruction (PR) Filter Banks Tree-structured filter banks Two level maximally decimated tree-structured filter banks If the QMF bank is alias free or PR systems, then so is the whole system. Different QMF banks can be used at different levels. Different tree-strictures (non-binary tree-structure) are also used.

Perfect Reconstruction (PR) Filter Banks Tree-structured filter banks Equivalent form

Perfect Reconstruction (PR) Filter Banks Tree-structured filter banks

Perfect Reconstruction (PR) Filter Banks Tree-structured filter banks

Perfect Reconstruction (PR) Filter Banks A Tree-structured filter bank having subbands Draw a tree-structured QMF bank of which the equivalent analysis filter bank has the above frequency responses. Subband coding for data compression Each band signal is quantized differently to compress the data. Data loss should be limited to a certain tolerable level. MPEG standard : audio compression with 32 subbands

Perfect Reconstruction (PR) Filter Banks Octave-Band Filter banks Redraw it so that it looks more tree-structured !!

Uniform DFT filter bank DFT matrix : matrix with : Conjugate of DFT matrix

Uniform DFT filter bank Let's define a set of filters (analysis filters) : : rectangular window

Uniform DFT filter bank Properties of the uniform DFT filter bank All filters , have the same length are obtained by shifting the prototype filter Right shift by due to the use of conjugate DFT matrix Mag response of is a sinc finction Too high sidelobe: -13.5 dB unsatisfactory for subband processing : complex filter Spectrum analyzer (or moving DFT)

Uniform DFT filter bank Properties of the uniform DFT filter bank Spectrum analyzer (or moving DFT) center frequency = Frequency resolution is not good. What if we use a better window to improve the frequency resolution ? This is equivalent to Analysis filter bank / Conjugate DFT matrix cascade Problems : Filter length limited to , RECTANGULAR WINDOW

Uniform DFT filter bank Output of the DFT matrix in the synthesis bank Since We will get Maximally decimated version: Decimated DFT filter bank : Perfect reconstruction Input vector to the conjugate DFT matrix Output: - Same as before - Perfect reconstruction

Uniform DFT filter bank Polyphase implementation of DFT filter bank DFT filter bank : Filter bank model in the previous section Consider an arbitrary length(N) filter with better characteristics : Polyphase representation (length

Uniform DFT filter bank Polyphase implementation of DFT filter bank Decimated DFT filter bank 2008.9 EEE4176 Applications of Digital Signal Processing 26

Uniform DFT filter bank Polyphase implementation of DFT filter bank Decimated DFT filter bank complex operations / unit sample time In direct structure, Can we make it have perfect reconstruction property ?